In three-space, if the angles between three unit vectors from the origin are known and the first two unit vectors are determined, how can I determine the third vector? This is a repost of my soccer ball (truncated icosahedron) question from a few days ago. I calculated the angles incorrectly in the previous post. Perhaps that is why it stumped everyone.

The first two vectors are in the XY plane at 126ÃÂÃÂº and 234ÃÂÃÂº from the +X axis. A third is along the +X axis at some angle above the XY plane. It is 120ÃÂÃÂº from either of the other vectors. What is the angle above the XY plane of the third vector?